Milind A. Sohoni

Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems

We suggest an approach based on geometric invariant theory to the fundamental lower bound problems in complexity theory concerning formula and circuit size. Specifically, we introduce the notion of a partially stable point in a reductive-group representation, which generalizes the notion of stability in geometric invariant theory due to Mumford [Geometric Invariant Theory, Springer-Verlag, Berlin, 1965]. Then we reduce fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points. We also suggest an approach to tackle the latter class of problems via construction of explicit obstructions.

A theory of regular MSC languages

Message sequence charts (MSCs) are an attractive visual formalism widely used to capture system requirements during the early design stages in domains such as telecommunication software. It is fruitful to have mechanisms for specifying and reasoning about collections of MSCs so that errors can be detected even at the requirements level. We propose, accordingly, a notion of regularity for collections of MSCs and explore its basic properties. In particular, we provide an automata-theoretic characterization of regular MSC languages in terms of finite-state distributed automata called bounded message-passing automata. These automata consist of a set of sequential processes that communicate with each other by sending and receiving messages over bounded FIFO channels. We also provide a logical characterization in terms of a natural monadic second-order logic interpreted over MSCs. A commonly used technique to generate a collection of MSCs is to use a hierarchical message sequence chart (HMSC). We show that the class of languages arising from the so-called bounded HMSCs constitute a proper subclass of the class of regular MSC languages. In fact, we characterize the bounded HMSC languages as the subclass of regular MSC languages that are finitely generated.

Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties

In [K. D. Mulmuley and M. Sohoni, SIAM J. Comput., 31 (2001), pp. 496-526], henceforth referred to as Part I, we suggested an approach to the

Synthesizing Distributed Finite-State Systems from MSCs

P

A graph-based framework for feature recognition

vs.

Keeping track of the latest gossip in a distributed system

NP

Removal of blends from boundary representation models

and related lower bound problems in complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the

Blend recognition algorithm and applications

NP \not \subseteq P

Geometric Complexity Theory, P vs. NP and Explicit Obstructions

conjecture to the problem of showing that a variety associated with the complexity class

Reconstruction of feature volumes and feature suppression

NP